Topics in Analysis, Examples Sheet 1. W. T. G. Michaelmas 2004

1. Let X be a non-compact subset of ℝ^{2}. Prove that there is a continuous unbounded function from X to ℝ.

2. Prove that a function f from a metric space X to a metric space Y is continuous if and only if f^{−1}(A) is closed in X for every closed set A ⊂ Y.

3. Let K be a compact subset of ℝ^{2} and let F be a closed subset of ℝ^{2} such that K ∩F = ∅. Give two proofs that there exists δ > 0 such that d(x, y) ⩾ δ for every x ∈ K and every y ∈ F, one proof directly from the compactness of K and the other using sequential compactness instead.

4. Let I, J and K be three arcs, each making an angle of 2π/3, whose union is the whole of the unit circle (as in lectures). Let A, B and C be open subsets of the closed unit disc D with I ⊂ A, J ⊂ B and K ⊂ C. Suppose that A ∪ B ∪ C is the whole of D. Deduce that A ∩ B ∩ C is non-empty from the corresponding result about closed sets. [Hint: One way to do it is to start by defining F ⊂ A to be the set of all x such that d(x, A^{c}) ⩾ max{d(x, B^{c}), d(x, C^{c})}, and similarly for sets G ⊂ B and H ⊂ C.]

5. Let f be a continuous function from the open unit disc to itself. Must it have a fixed point? Does your answer change if f is a surjection?

6. Let X and Y be metric spaces, let K be a compact subset of X and let f : X → Y be continuous. Prove that f(K) is closed.

7. Let f be a continuous function from a compact metric space X to itself, and suppose that d(f(x), f(y)) = d(x, y) for every x, y ∈ X. Prove that f is a surjection. [Hint: if not, pick x ∉ f(X), show that there is a ball about x that misses f(X) and consider the sequence x, f(x), f(f(x)), . . ..]

8. Let x be a point in the closed unit disc D, let K be a compact subset of D not containing x and define a map f_{x} : K → ∂D as follows. Given y ∈ K, take the line from x to y and extend it (in the x-to-y direction) until it first hits the boundary. Call this point f_{x}(y). Prove that f_{x} is a continuous function.

9. Let g : D → D be a continuous function and let x ∈ D be a point such that x ≠ g(x). Let h(x) = f_{x}(g(x)), where f_{x} is defined as in question 8 (on some open set containing x). Prove that h is continuous at x.

10. Let C[0, 1] be the metric space consisting of all continuous functions f : [0, 1] → ℝ, with the distance d(f, g) between two functions f and g defined to be the supremum of |f(x) − g(x)| over all x ∈ [0, 1].

(i) Explain why this supremum is in fact a maximum.

(ii) Let X consist of all functions f in C[0, 1] that take values in [0, 1]. Prove that X is not a sequentially compact subset of C[0, 1].

(iii) This shows that X is not compact. Prove the same result by exhibiting an open cover of X that has no finite subcover.

11. Prove that every compact metric space is complete. The diameter of a metric space X is the supremum of d(x, y) over all points x, y ∈ X. Give an example of a metric space that is complete and has diameter 1 but is not compact.

12. Let A be a 3 × 3-matrix with positive entries. Use Brouwer’s fixed-point theorem to prove that A has an eigenvector with positive entries. [Hint: use A to define a map from T to itself, where T is the triangle with vertices (1, 0, 0), (0, 1, 0) and (0, 0, 1).]